Semi-implicit Euler scheme for generalized Newtonian fluids

Rheological behavior of certain non-Newtonian fluids in engineering sciences is often modeled by power law ansatzes with p <= 2. So far, existing numerical analysis for local strong solutions studies a fully implicit time discretization and find only restricted ranges of admissible p's for corresponding error estimates [ A. Prohl and M. Ruzicka, SIAM J. Numer. Anal., 39 ( 2001), pp. 214 - 249]; different nonlinear stabilization strategies which allow a corresponding error analysis for smaller p's are examined in [ L. Diening, Theoretical and Numerical Results for Electrorheological Fluids, Ph. D. thesis, University of Freiburg, Freiburg, Germany, 2002] and [ L. Diening, A. Prohl, and M. Ruzicka, in Nonlinear Problems in Mathematical Physics and Related Topics, II, Kluwer/Plenum, New York, 2002, pp. 89 - 118]. In the present paper, a semi-implicit time discretization scheme is proposed, and error estimates apply to the extended range p is an element of ( 3/2, 2]. The key analytical tool is a new Gronwall-type inequality.